Math and stuff
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The Jacobson radical and Nakayama's lemma
Proposition Let a be an ideal of a ring A, and let S=1+a. Show that S−1a is contained in the Jacobson radical of S−1A. Use this result and Nakayama’s lemma to give a proof of (2.5)[Atiyah] which does not depend on determinants. Solution 0∈a,...
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If f has a zero of multiplicity m at a, then 1/f has a pole of order m at a
Proposition Suppose that f has a zero of multiplicity m at a. Explain why 1/f has a pole of order m at a. Solution f has a zero of multiplicity m at a. By Theorem 8.14[A first course in complex analysis], this means that there exists a holomorphic function g...
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Poles and removable singularities
Proposition Find the poles or removable singularities of the following functions and determine their orders: (z2+1)−3(z−1)−4. zcot(z). z−5sinz. z1−exp(z). 11−exp(z). Solution 1 The order of the pole at i is 3. The order of the pole at 1 is 4. 2...
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If f has an essential singularity at z0, then so does 1/f
Proposition Show that if f has an essential singularity at z0 then 1f also has an essential singularity at z0. Solution Suppose 1/f does not have an essential singularity at z0. Since f has an isolated singularity at z0, 1/f has an isolated singularity at z0. By Proposition 9.5[a first...
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Cross products and determinants
Proposition If w1,⋯,wn−1∈Rn, show that |w1×⋯×wn−1|=√det(gij), where gij=⟨wi,wj⟩. Solution Suppose w1,⋯,wn−1 are linearly dependent. Then w1×⋯×wn−1=0 because $\ev{w_1 \times...